88
Chapter 5. The Derivative
Chapter 6
Sequences and Series of
Functions
6.1
Discussion: Branching Processes
6.2
Uniform Convergence of a Sequence
of Functions
Exercise 6.2.1. (a) By dividing the numerator and denominator by n, we can
compute
nx
x
1
lim f
5.4. A Continuous Nowhere-Dierentiable Function
85
Turning our attention to the dierence quotient, we get
g(xm ) g(0)
=
xm 0
Pm
m
X
1/2m
=
1 = m + 1.
1/2m
n=0
n=0
Because this quantity increases without bound, it is impossible for limm!1 g(xm )/xm
to exis
94
Chapter 6. Sequences and Series of Functions
and choose N2 so that
|gn gm | < /2
for all m, n N2 .
Letting N = maxcfw_N1 , N2 we see that
|(fn + gn ) (fm + gm )| |fn fm | + |gn gm |
<
+ =
2 2
for all m, n N . It follows from the Cauchy Criterion for U
100
Chapter 6. Sequences and Series of Functions
Because this holds for all m N , we can set m = n 1 to get that
|fn (x)| <
whenever n > N .
This proves gn ! 0 uniformly.
Exercise 6.4.2. The key idea is to use the Cauchy criterion for convergence of
aPse
5.3. The Mean Value Theorem
79
closer and closer to zero. To see this explicitly, set tn = 1/(2n) and xn = 0.
Then observe that |xn tn | ! 0 while
g2 (xn ) g2 (tn )
0
g2 (tn ) = |tn sin(1/tn ) + cos(1/tn ) 2tn sin(1/tn )|
xn t n
= |cos(1/tn ) tn sin(1/t
103
6.5. Power Series
P
Exercise 6.5.6. Assume
an xn converges pointwise on the compact set K.
Because K is compact, there exist points x0 , x1 2 K satisfying
x0 x x1
for all x 2 K.
At this point we need to consider a few dierent cases. If x0 > 0 then K [
6.2. Uniform Convergence of a Sequence of Functions
91
Exercise 6.2.5. (a) Taking the limit for each fixed value of x we find that fn (x)
converges pointwise to
1 if x 6= 0
f (x) =
0 if x = 0.
Each of the functions fn is continuous, but the limit function
106
Chapter 6. Sequences and Series of Functions
Exercise 6.6.3. The key idea is to take the derivative of each side of equation
(2) using a term-by-term approach for the series on the right (this is justified
by Theorem 6.5.7). Setting x = 0 after n deri
6.3. Uniform Convergence and Dierentiation
97
Letting N = maxcfw_N1 , N2 , . . . , Nm produces the desired N .
Note that if the set cfw_r1 , r2 , . . . , rm were infinite then N would be the maximum of an infinite set which is problematic to say the lea
82
Chapter 5. The Derivative
() For the other direction we use the Mean Value Theorem. Here we are
assuming f 0 (x) 0 on (a, b) and we are asked to prove that f is increasing.
Given x < y, it follows from MVT that
f 0 (c) =
f (y) f (x)
yx
for some point c